Theorem bifwd-P2.5a.2AC.SH 114

Description: Another Deductive form of bifwd-P2.5a 111.

Hypothesis

Ref	Expression
bifwd-P2.5a.2AC.SH.1	⊢ (𝛾₁ → (𝛾₂ → (𝜑 ↔ 𝜓)))

Assertion

Ref	Expression
bifwd-P2.5a.2AC.SH	⊢ (𝛾₁ → (𝛾₂ → (𝜑 → 𝜓)))

Proof of Theorem bifwd-P2.5a.2AC.SH

Step	Hyp	Ref	Expression
1		bifwd-P2.5a.2AC.SH.1	. 2 ⊢ (𝛾₁ → (𝛾₂ → (𝜑 ↔ 𝜓)))
2		bifwd-P2.5a 111	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	2	axL1.SH 30	. . . 4 ⊢ (𝛾₂ → ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)))
4	3	axL1.SH 30	. . 3 ⊢ (𝛾₁ → (𝛾₂ → ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))))
5	4	rcp-FR2.SH 42	. 2 ⊢ ((𝛾₁ → (𝛾₂ → (𝜑 ↔ 𝜓))) → (𝛾₁ → (𝛾₂ → (𝜑 → 𝜓))))
6	1, 5	ax-MP 14	1 ⊢ (𝛾₁ → (𝛾₂ → (𝜑 → 𝜓)))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107

This theorem is referenced by:  bitrns-P2.6c  126